Three-dimensional representation of the pure electric-dipole and the mixed first hyperpolarizabilities: The modified unit sphere representation

In this work, the theory of the modified unit sphere representation (mUSR) has been proposed as a computational tool suitable for the three-dimensional representation of the pure electric-dipole [${\beta }_{\lambda \mu \nu }(-2\omega ;\omega ,\omega )$] as well as of the mixed electric-dipole/magnetic-dipole [${\hspace{0.17em}}^{\alpha }{J}_{\lambda \mu \nu }(-2\omega ;\omega ,\omega )$ and ${\hspace{0.17em}}^{\beta }{J}_{\lambda \mu \nu }(-2\omega ;\omega ,\omega )$] or electric-dipole/electric-quadrupole [${\hspace{0.17em}}^{\alpha }{K}_{\lambda \mu \nu o}(-2\omega ;\omega ,\omega )$ and ${\hspace{0.17em}}^{\beta }{K}_{\lambda \mu \nu o}(-2\omega ;\omega ,\omega )$] first hyperpolarizabilities. These five quantities are Cartesian tensors and they are responsible for the chiral signal in the chiroptical version of the hyper-Rayleigh scattering (HRS) spectroscopy, namely the HRS optical activity (HRS-OA) spectroscopy. For the first time, for each hyperpolarizability, alongside with the three-dimensional representation of the whole (i.e., reducible) Cartesian tensors, the mUSRs are developed for each of the irreducible Cartesian tensors (ICTs) that constitute them. This scheme has been applied to a series of three (chiral) hexahelicene molecules containing different degrees of electron-withdrawing (quinone) groups and characterized by the same (positive) handedness. For these molecules, the mUSR shows that, upon substitution, the most remarkable qualitative and semi-quantitative (enhancement of the molecular responses) effects are obtained for the pure electric-dipole and for the mixed electric-dipole/magnetic-dipole hyperpolarizabilities.